Here’s a site offering a free toothpaste sample: http://tryahsensitive.com.
It’s for Arm & Hammer Sensitive toothpaste. The site says you can request one sample per U.S. household while supplies last. It doesn’t say how big the sample is.
While it’s great when toothpaste samples come in travel size tubes, I’ve also discovered that the even smaller toothpaste packets are terrific too. They usually contain enough paste for several brushings and are smaller than a ketchup packet, making them the perfect size to stick in an overnight bag for a short trip.